A Generic Proximal Algorithm for Convex Optimization—Application to Total Variation Minimization

Abstract

We propose new optimization algorithms to minimize a sum of convex functions, which may be smooth or not and composed or not with linear operators. This generic formulation encompasses various forms of regularized inverse problems in imaging. The proposed algorithms proceed by splitting: the gradient or proximal operators of the functions are called individually, without inner loop or linear system to solve at each iteration. The algorithms are easy to implement and have proven convergence to an exact solution. The classical Douglas-Rachford and forward-backward splitting methods, as well as the recent and efficient algorithm of Chambolle-Pock, are recovered as particular cases. The application to inverse imaging problems regularized by the total variation is detailed.